1 Monte Carlo standard error

1.1 One-step selection mechanism

1.1.1 \(k=10\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00397 0.02951 0.00893 0.00766 0.00792
4PSM 0.00747 0.09246 0.00882 0.00471 0.00901
CHE 0.00196 0.00180 0.00660 0.00248 0.00378
CHE-ISCW 0.00212 0.00193 0.00580 0.00626 0.00513
MV_EK 0.00496 0.00505 0.00461 0.03067 0.00723
MV_PEESE 0.00321 0.00299 0.00476 0.01833 0.00733
MV_PET 0.00566 0.00535 0.00427 0.03287 0.00743
MV_PET-PEESE 0.00502 0.00507 0.00473 0.03399 0.00775
MV_WAAP 0.00233 0.00221 0.00562 0.01589 0.00642
MV_WILS 0.00262 0.00229 0.00831 0.00750 0.00559
TF 0.00264 0.00256 0.00947 0.00061 0.00319
UNI_EK 0.00515 0.00531 0.00521 0.02908 0.00755
UNI_FE 0.00233 0.00219 0.00659 0.00619 0.00527
UNI_PEESE 0.00344 0.00327 0.00519 0.01961 0.00753
UNI_PET 0.00606 0.00583 0.00487 0.03107 0.00766
UNI_PET-PEESE 0.00541 0.00554 0.00521 0.03209 0.00791
UNI_RE 0.00221 0.00211 0.00675 0.00441 0.00444
UNI_RE-ISW 0.00233 0.00219 0.00612 0.00987 0.00602
UNI_WAAP 0.00254 0.00241 0.00592 0.01713 0.00703
UNI_WILS 0.00296 0.00257 0.00855 0.01042 0.00618
p-uniform 0.07733 2.25425 0.00946 0.04081 0.00420
p-uniform* 0.52835 11.73175 0.00875 0.00255 0.00663

1.1.2 \(k=30\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00178 0.00330 0.00901 0.00148 0.00462
4PSM 0.00241 0.01341 0.00843 0.00203 0.00746
CHE 0.00111 0.00103 0.00675 0.00073 0.00215
CHE-ISCW 0.00122 0.00112 0.00657 0.00122 0.00249
MV_EK 0.00274 0.00250 0.00559 0.00437 0.00566
MV_PEESE 0.00174 0.00157 0.00620 0.00230 0.00354
MV_PET 0.00307 0.00274 0.00555 0.00461 0.00593
MV_PET-PEESE 0.00244 0.00221 0.00612 0.00470 0.00549
MV_WAAP 0.00145 0.00148 0.00657 0.00712 0.00378
MV_WILS 0.00215 0.00195 0.00708 0.00290 0.00526
TF 0.00149 0.00140 0.00827 0.00019 0.00211
UNI_EK 0.00308 0.00289 0.00611 0.00683 0.00620
UNI_FE 0.00141 0.00132 0.00680 0.00204 0.00271
UNI_PEESE 0.00198 0.00184 0.00655 0.00404 0.00406
UNI_PET 0.00348 0.00318 0.00598 0.00736 0.00679
UNI_PET-PEESE 0.00285 0.00270 0.00656 0.00743 0.00622
UNI_RE 0.00128 0.00121 0.00667 0.00142 0.00230
UNI_RE-ISW 0.00141 0.00132 0.00660 0.00237 0.00282
UNI_WAAP 0.00165 0.00164 0.00685 0.00754 0.00408
UNI_WILS 0.00251 0.00216 0.00758 0.00442 0.00599
p-uniform 0.01879 0.52100 0.00768 0.00562 0.00224
p-uniform* 0.00920 0.33734 0.00886 0.00103 0.00413

1.1.3 \(k=60\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00123 0.00113 0.00904 0.00058 0.00329
4PSM 0.00149 0.00168 0.00834 0.00105 0.00568
CHE 0.00078 0.00073 0.00627 0.00035 0.00165
CHE-ISCW 0.00086 0.00080 0.00640 0.00057 0.00186
MV_EK 0.00192 0.00172 0.00607 0.00181 0.00417
MV_PEESE 0.00122 0.00111 0.00656 0.00098 0.00244
MV_PET 0.00215 0.00192 0.00623 0.00189 0.00424
MV_PET-PEESE 0.00159 0.00138 0.00659 0.00185 0.00389
MV_WAAP 0.00107 0.00112 0.00672 0.00451 0.00307
MV_WILS 0.00174 0.00156 0.00654 0.00164 0.00502
TF 0.00104 0.00097 0.00744 0.00009 0.00174
UNI_EK 0.00219 0.00200 0.00662 0.00305 0.00466
UNI_FE 0.00101 0.00095 0.00640 0.00108 0.00195
UNI_PEESE 0.00141 0.00131 0.00667 0.00183 0.00263
UNI_PET 0.00247 0.00224 0.00668 0.00325 0.00483
UNI_PET-PEESE 0.00185 0.00164 0.00686 0.00296 0.00431
UNI_RE 0.00091 0.00086 0.00617 0.00074 0.00169
UNI_RE-ISW 0.00101 0.00095 0.00631 0.00115 0.00196
UNI_WAAP 0.00123 0.00124 0.00682 0.00484 0.00317
UNI_WILS 0.00205 0.00174 0.00687 0.00244 0.00570
p-uniform 0.00469 0.11903 0.00657 0.00459 0.00147
p-uniform* 0.00120 0.00108 0.00892 0.00055 0.00311

1.1.4 \(k=100\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00095 0.00085 0.00907 0.00034 0.00270
4PSM 0.00114 0.00114 0.00839 0.00064 0.00448
CHE 0.00060 0.00057 0.00577 0.00021 0.00138
CHE-ISCW 0.00067 0.00062 0.00608 0.00033 0.00155
MV_EK 0.00148 0.00132 0.00637 0.00103 0.00318
MV_PEESE 0.00094 0.00086 0.00654 0.00056 0.00199
MV_PET 0.00165 0.00149 0.00662 0.00107 0.00321
MV_PET-PEESE 0.00117 0.00101 0.00672 0.00102 0.00298
MV_WAAP 0.00085 0.00090 0.00662 0.00316 0.00258
MV_WILS 0.00147 0.00131 0.00550 0.00111 0.00469
TF 0.00080 0.00075 0.00680 0.00006 0.00152
UNI_EK 0.00170 0.00154 0.00686 0.00178 0.00348
UNI_FE 0.00079 0.00074 0.00602 0.00068 0.00157
UNI_PEESE 0.00109 0.00102 0.00645 0.00109 0.00204
UNI_PET 0.00192 0.00175 0.00702 0.00190 0.00352
UNI_PET-PEESE 0.00135 0.00120 0.00675 0.00163 0.00319
UNI_RE 0.00071 0.00067 0.00570 0.00045 0.00140
UNI_RE-ISW 0.00079 0.00074 0.00597 0.00070 0.00158
UNI_WAAP 0.00098 0.00101 0.00656 0.00343 0.00264
UNI_WILS 0.00175 0.00147 0.00578 0.00158 0.00530
p-uniform 0.00197 0.03545 0.00583 0.00734 0.00113
p-uniform* 0.00093 0.00082 0.00896 0.00032 0.00263

1.2 Two-step selection mechanism

1.2.1 \(k=10\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00459 0.04891 0.00958 0.00591 0.00735
4PSM 0.00924 0.11167 0.00843 0.00669 0.00843
CHE 0.00186 0.00175 0.00727 0.00245 0.00368
CHE-ISCW 0.00202 0.00187 0.00653 0.00601 0.00527
MV_EK 0.00470 0.00482 0.00485 0.02841 0.00751
MV_PEESE 0.00305 0.00286 0.00516 0.01708 0.00766
MV_PET 0.00539 0.00515 0.00440 0.03066 0.00761
MV_PET-PEESE 0.00482 0.00490 0.00510 0.03177 0.00807
MV_WAAP 0.00223 0.00212 0.00636 0.01560 0.00679
MV_WILS 0.00252 0.00220 0.00893 0.00716 0.00573
TF 0.00247 0.00236 0.00918 0.00061 0.00267
UNI_EK 0.00482 0.00492 0.00549 0.02658 0.00783
UNI_FE 0.00218 0.00210 0.00724 0.00584 0.00535
UNI_PEESE 0.00322 0.00311 0.00557 0.01803 0.00791
UNI_PET 0.00569 0.00554 0.00500 0.02861 0.00783
UNI_PET-PEESE 0.00511 0.00526 0.00556 0.02962 0.00824
UNI_RE 0.00207 0.00204 0.00738 0.00423 0.00424
UNI_RE-ISW 0.00218 0.00210 0.00687 0.00911 0.00627
UNI_WAAP 0.00239 0.00229 0.00664 0.01664 0.00748
UNI_WILS 0.00277 0.00245 0.00900 0.00944 0.00630
p-uniform 0.05021 1.40334 0.00941 0.03301 0.00409
p-uniform* 0.80565 15.94497 0.00906 0.00252 0.00657

1.2.2 \(k=30\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00174 0.00323 0.01020 0.00156 0.00462
4PSM 0.00295 0.02316 0.00803 0.00438 0.00804
CHE 0.00106 0.00101 0.00712 0.00072 0.00216
CHE-ISCW 0.00116 0.00110 0.00710 0.00121 0.00258
MV_EK 0.00261 0.00240 0.00595 0.00421 0.00585
MV_PEESE 0.00166 0.00152 0.00673 0.00224 0.00355
MV_PET 0.00294 0.00264 0.00588 0.00446 0.00614
MV_PET-PEESE 0.00235 0.00211 0.00665 0.00447 0.00571
MV_WAAP 0.00139 0.00138 0.00729 0.00695 0.00407
MV_WILS 0.00205 0.00186 0.00725 0.00290 0.00525
TF 0.00140 0.00134 0.00756 0.00020 0.00154
UNI_EK 0.00286 0.00271 0.00652 0.00627 0.00642
UNI_FE 0.00131 0.00128 0.00723 0.00191 0.00251
UNI_PEESE 0.00185 0.00176 0.00705 0.00377 0.00394
UNI_PET 0.00325 0.00301 0.00629 0.00689 0.00706
UNI_PET-PEESE 0.00268 0.00252 0.00709 0.00688 0.00651
UNI_RE 0.00120 0.00117 0.00707 0.00135 0.00211
UNI_RE-ISW 0.00131 0.00128 0.00713 0.00221 0.00267
UNI_WAAP 0.00155 0.00152 0.00757 0.00727 0.00427
UNI_WILS 0.00235 0.00204 0.00763 0.00393 0.00576
p-uniform 0.00858 0.20254 0.00757 0.00680 0.00218
p-uniform* 0.00165 0.00190 0.00993 0.00103 0.00415

1.2.3 \(k=60\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00120 0.00116 0.01027 0.00061 0.00336
4PSM 0.00165 0.00337 0.00797 0.00152 0.00636
CHE 0.00074 0.00071 0.00650 0.00035 0.00163
CHE-ISCW 0.00082 0.00078 0.00677 0.00056 0.00189
MV_EK 0.00183 0.00165 0.00646 0.00177 0.00417
MV_PEESE 0.00116 0.00107 0.00704 0.00097 0.00247
MV_PET 0.00205 0.00184 0.00662 0.00186 0.00427
MV_PET-PEESE 0.00153 0.00130 0.00707 0.00177 0.00398
MV_WAAP 0.00104 0.00104 0.00729 0.00462 0.00329
MV_WILS 0.00163 0.00147 0.00702 0.00165 0.00496
TF 0.00098 0.00093 0.00631 0.00010 0.00094
UNI_EK 0.00204 0.00189 0.00704 0.00286 0.00458
UNI_FE 0.00094 0.00091 0.00665 0.00102 0.00175
UNI_PEESE 0.00131 0.00125 0.00711 0.00173 0.00246
UNI_PET 0.00231 0.00213 0.00706 0.00309 0.00481
UNI_PET-PEESE 0.00172 0.00150 0.00731 0.00264 0.00426
UNI_RE 0.00085 0.00083 0.00632 0.00070 0.00150
UNI_RE-ISW 0.00094 0.00091 0.00663 0.00108 0.00179
UNI_WAAP 0.00116 0.00115 0.00730 0.00469 0.00325
UNI_WILS 0.00191 0.00164 0.00737 0.00218 0.00543
p-uniform 0.00625 0.19220 0.00650 0.02524 0.00144
p-uniform* 0.00116 0.00108 0.01002 0.00058 0.00314

1.2.4 \(k=100\)

method bias_mcse_m rmse_mcse_m cover_mcse_m width_mcse_m rej_mcse_m
3PSM 0.00092 0.00087 0.00989 0.00036 0.00254
4PSM 0.00124 0.00151 0.00788 0.00096 0.00512
CHE 0.00058 0.00056 0.00579 0.00021 0.00126
CHE-ISCW 0.00064 0.00061 0.00628 0.00033 0.00151
MV_EK 0.00141 0.00127 0.00675 0.00101 0.00313
MV_PEESE 0.00090 0.00084 0.00691 0.00056 0.00208
MV_PET 0.00158 0.00143 0.00700 0.00106 0.00318
MV_PET-PEESE 0.00112 0.00096 0.00709 0.00098 0.00303
MV_WAAP 0.00083 0.00085 0.00698 0.00333 0.00274
MV_WILS 0.00135 0.00120 0.00635 0.00109 0.00462
TF 0.00075 0.00072 0.00533 0.00006 0.00052
UNI_EK 0.00158 0.00146 0.00722 0.00168 0.00324
UNI_FE 0.00073 0.00071 0.00602 0.00064 0.00126
UNI_PEESE 0.00102 0.00097 0.00684 0.00103 0.00199
UNI_PET 0.00178 0.00165 0.00736 0.00181 0.00331
UNI_PET-PEESE 0.00125 0.00110 0.00709 0.00145 0.00305
UNI_RE 0.00066 0.00064 0.00551 0.00044 0.00102
UNI_RE-ISW 0.00073 0.00071 0.00601 0.00066 0.00128
UNI_WAAP 0.00093 0.00092 0.00676 0.00349 0.00254
UNI_WILS 0.00162 0.00135 0.00665 0.00142 0.00500
p-uniform 0.00125 0.00604 0.00577 0.00216 0.00108
p-uniform* 0.00090 0.00084 0.00974 0.00034 0.00238

2 Convergence rates

2.1 One-step selection mechanism

2.1.1 \(k=10\)

sel_mechanism k method mean_convergence min_convergence max_convergence
One-step selection 10 3PSM 0.99411 0.9695 1
One-step selection 10 4PSM 0.98485 0.9025 1
One-step selection 10 CHE 1.00000 1.0000 1
One-step selection 10 CHE-ISCW 1.00000 1.0000 1
One-step selection 10 MV_EK 0.99999 0.9995 1
One-step selection 10 MV_PEESE 1.00000 1.0000 1
One-step selection 10 MV_PET 1.00000 1.0000 1
One-step selection 10 MV_PET-PEESE 1.00000 1.0000 1
One-step selection 10 MV_WAAP 1.00000 1.0000 1
One-step selection 10 MV_WILS 0.99999 0.9995 1
One-step selection 10 TF 1.00000 1.0000 1
One-step selection 10 UNI_EK 0.99999 0.9990 1
One-step selection 10 UNI_FE 1.00000 1.0000 1
One-step selection 10 UNI_PEESE 1.00000 1.0000 1
One-step selection 10 UNI_PET 1.00000 1.0000 1
One-step selection 10 UNI_PET-PEESE 1.00000 1.0000 1
One-step selection 10 UNI_RE 0.99905 0.9920 1
One-step selection 10 UNI_RE-ISW 0.99905 0.9920 1
One-step selection 10 UNI_WAAP 1.00000 1.0000 1
One-step selection 10 UNI_WILS 1.00000 1.0000 1
One-step selection 10 p-uniform 0.97173 0.4355 1
One-step selection 10 p-uniform* 1.00000 1.0000 1

2.1.2 \(k=30\)

sel_mechanism k method mean_convergence min_convergence max_convergence
One-step selection 30 3PSM 0.99963 0.9930 1
One-step selection 30 4PSM 0.99936 0.9945 1
One-step selection 30 CHE 1.00000 1.0000 1
One-step selection 30 CHE-ISCW 1.00000 1.0000 1
One-step selection 30 MV_EK 1.00000 1.0000 1
One-step selection 30 MV_PEESE 1.00000 1.0000 1
One-step selection 30 MV_PET 1.00000 1.0000 1
One-step selection 30 MV_PET-PEESE 1.00000 1.0000 1
One-step selection 30 MV_WAAP 1.00000 1.0000 1
One-step selection 30 MV_WILS 1.00000 0.9995 1
One-step selection 30 TF 1.00000 1.0000 1
One-step selection 30 UNI_EK 1.00000 1.0000 1
One-step selection 30 UNI_FE 1.00000 1.0000 1
One-step selection 30 UNI_PEESE 1.00000 1.0000 1
One-step selection 30 UNI_PET 1.00000 1.0000 1
One-step selection 30 UNI_PET-PEESE 1.00000 1.0000 1
One-step selection 30 UNI_RE 0.99998 0.9990 1
One-step selection 30 UNI_RE-ISW 0.99998 0.9990 1
One-step selection 30 UNI_WAAP 1.00000 1.0000 1
One-step selection 30 UNI_WILS 1.00000 1.0000 1
One-step selection 30 p-uniform 0.99636 0.8025 1
One-step selection 30 p-uniform* 1.00000 1.0000 1

2.1.3 \(k=60\)

sel_mechanism k method mean_convergence min_convergence max_convergence
One-step selection 60 3PSM 0.99987 0.9935 1
One-step selection 60 4PSM 0.99986 0.9960 1
One-step selection 60 CHE 1.00000 1.0000 1
One-step selection 60 CHE-ISCW 1.00000 1.0000 1
One-step selection 60 MV_EK 1.00000 1.0000 1
One-step selection 60 MV_PEESE 1.00000 1.0000 1
One-step selection 60 MV_PET 1.00000 1.0000 1
One-step selection 60 MV_PET-PEESE 1.00000 1.0000 1
One-step selection 60 MV_WAAP 1.00000 1.0000 1
One-step selection 60 MV_WILS 1.00000 0.9995 1
One-step selection 60 TF 1.00000 1.0000 1
One-step selection 60 UNI_EK 1.00000 1.0000 1
One-step selection 60 UNI_FE 1.00000 1.0000 1
One-step selection 60 UNI_PEESE 1.00000 1.0000 1
One-step selection 60 UNI_PET 1.00000 1.0000 1
One-step selection 60 UNI_PET-PEESE 1.00000 1.0000 1
One-step selection 60 UNI_RE 1.00000 1.0000 1
One-step selection 60 UNI_RE-ISW 1.00000 1.0000 1
One-step selection 60 UNI_WAAP 1.00000 1.0000 1
One-step selection 60 UNI_WILS 1.00000 1.0000 1
One-step selection 60 p-uniform 0.99963 0.9655 1
One-step selection 60 p-uniform* 1.00000 1.0000 1

2.1.4 \(k=100\)

sel_mechanism k method mean_convergence min_convergence max_convergence
One-step selection 100 3PSM 0.99997 0.9980 1
One-step selection 100 4PSM 0.99995 0.9985 1
One-step selection 100 CHE 1.00000 1.0000 1
One-step selection 100 CHE-ISCW 1.00000 1.0000 1
One-step selection 100 MV_EK 1.00000 1.0000 1
One-step selection 100 MV_PEESE 1.00000 1.0000 1
One-step selection 100 MV_PET 1.00000 1.0000 1
One-step selection 100 MV_PET-PEESE 1.00000 1.0000 1
One-step selection 100 MV_WAAP 1.00000 1.0000 1
One-step selection 100 MV_WILS 1.00000 0.9995 1
One-step selection 100 TF 1.00000 1.0000 1
One-step selection 100 UNI_EK 1.00000 1.0000 1
One-step selection 100 UNI_FE 1.00000 1.0000 1
One-step selection 100 UNI_PEESE 1.00000 1.0000 1
One-step selection 100 UNI_PET 1.00000 1.0000 1
One-step selection 100 UNI_PET-PEESE 1.00000 1.0000 1
One-step selection 100 UNI_RE 1.00000 1.0000 1
One-step selection 100 UNI_RE-ISW 1.00000 1.0000 1
One-step selection 100 UNI_WAAP 1.00000 1.0000 1
One-step selection 100 UNI_WILS 1.00000 1.0000 1
One-step selection 100 p-uniform 0.99997 0.9970 1
One-step selection 100 p-uniform* 1.00000 1.0000 1

2.2 Two-step selection mechanism

2.2.1 \(k=10\)

sel_mechanism k method mean_convergence min_convergence max_convergence
Two-step selection 10 3PSM 0.99325 0.9680 1
Two-step selection 10 4PSM 0.98180 0.8920 1
Two-step selection 10 CHE 1.00000 0.9995 1
Two-step selection 10 CHE-ISCW 1.00000 1.0000 1
Two-step selection 10 MV_EK 0.99999 0.9995 1
Two-step selection 10 MV_PEESE 1.00000 1.0000 1
Two-step selection 10 MV_PET 1.00000 1.0000 1
Two-step selection 10 MV_PET-PEESE 1.00000 1.0000 1
Two-step selection 10 MV_WAAP 1.00000 1.0000 1
Two-step selection 10 MV_WILS 0.99999 0.9995 1
Two-step selection 10 TF 1.00000 1.0000 1
Two-step selection 10 UNI_EK 1.00000 0.9995 1
Two-step selection 10 UNI_FE 1.00000 1.0000 1
Two-step selection 10 UNI_PEESE 1.00000 1.0000 1
Two-step selection 10 UNI_PET 1.00000 1.0000 1
Two-step selection 10 UNI_PET-PEESE 1.00000 1.0000 1
Two-step selection 10 UNI_RE 0.99888 0.9925 1
Two-step selection 10 UNI_RE-ISW 0.99888 0.9925 1
Two-step selection 10 UNI_WAAP 1.00000 1.0000 1
Two-step selection 10 UNI_WILS 1.00000 1.0000 1
Two-step selection 10 p-uniform 0.97642 0.4375 1
Two-step selection 10 p-uniform* 1.00000 1.0000 1

2.2.2 \(k=30\)

sel_mechanism k method mean_convergence min_convergence max_convergence
Two-step selection 30 3PSM 0.99959 0.9935 1
Two-step selection 30 4PSM 0.99911 0.9920 1
Two-step selection 30 CHE 1.00000 1.0000 1
Two-step selection 30 CHE-ISCW 1.00000 1.0000 1
Two-step selection 30 MV_EK 1.00000 1.0000 1
Two-step selection 30 MV_PEESE 1.00000 1.0000 1
Two-step selection 30 MV_PET 1.00000 1.0000 1
Two-step selection 30 MV_PET-PEESE 1.00000 1.0000 1
Two-step selection 30 MV_WAAP 1.00000 1.0000 1
Two-step selection 30 MV_WILS 1.00000 0.9995 1
Two-step selection 30 TF 1.00000 1.0000 1
Two-step selection 30 UNI_EK 1.00000 1.0000 1
Two-step selection 30 UNI_FE 1.00000 1.0000 1
Two-step selection 30 UNI_PEESE 1.00000 1.0000 1
Two-step selection 30 UNI_PET 1.00000 1.0000 1
Two-step selection 30 UNI_PET-PEESE 1.00000 1.0000 1
Two-step selection 30 UNI_RE 0.99997 0.9990 1
Two-step selection 30 UNI_RE-ISW 0.99997 0.9990 1
Two-step selection 30 UNI_WAAP 1.00000 1.0000 1
Two-step selection 30 UNI_WILS 1.00000 1.0000 1
Two-step selection 30 p-uniform 0.99726 0.8305 1
Two-step selection 30 p-uniform* 1.00000 1.0000 1

2.2.3 \(k=60\)

sel_mechanism k method mean_convergence min_convergence max_convergence
Two-step selection 60 3PSM 0.99991 0.9970 1
Two-step selection 60 4PSM 0.99987 0.9985 1
Two-step selection 60 CHE 1.00000 1.0000 1
Two-step selection 60 CHE-ISCW 1.00000 1.0000 1
Two-step selection 60 MV_EK 1.00000 1.0000 1
Two-step selection 60 MV_PEESE 1.00000 1.0000 1
Two-step selection 60 MV_PET 1.00000 1.0000 1
Two-step selection 60 MV_PET-PEESE 1.00000 1.0000 1
Two-step selection 60 MV_WAAP 1.00000 1.0000 1
Two-step selection 60 MV_WILS 1.00000 0.9995 1
Two-step selection 60 TF 1.00000 1.0000 1
Two-step selection 60 UNI_EK 1.00000 1.0000 1
Two-step selection 60 UNI_FE 1.00000 1.0000 1
Two-step selection 60 UNI_PEESE 1.00000 1.0000 1
Two-step selection 60 UNI_PET 1.00000 1.0000 1
Two-step selection 60 UNI_PET-PEESE 1.00000 1.0000 1
Two-step selection 60 UNI_RE 1.00000 1.0000 1
Two-step selection 60 UNI_RE-ISW 1.00000 1.0000 1
Two-step selection 60 UNI_WAAP 1.00000 1.0000 1
Two-step selection 60 UNI_WILS 1.00000 1.0000 1
Two-step selection 60 p-uniform 0.99975 0.9725 1
Two-step selection 60 p-uniform* 1.00000 1.0000 1

2.2.4 \(k=100\)

sel_mechanism k method mean_convergence min_convergence max_convergence
Two-step selection 100 3PSM 0.99998 0.9980 1
Two-step selection 100 4PSM 0.99995 0.9985 1
Two-step selection 100 CHE 1.00000 0.9995 1
Two-step selection 100 CHE-ISCW 1.00000 1.0000 1
Two-step selection 100 MV_EK 1.00000 1.0000 1
Two-step selection 100 MV_PEESE 1.00000 1.0000 1
Two-step selection 100 MV_PET 1.00000 1.0000 1
Two-step selection 100 MV_PET-PEESE 1.00000 1.0000 1
Two-step selection 100 MV_WAAP 1.00000 1.0000 1
Two-step selection 100 MV_WILS 1.00000 0.9995 1
Two-step selection 100 TF 1.00000 1.0000 1
Two-step selection 100 UNI_EK 1.00000 1.0000 1
Two-step selection 100 UNI_FE 1.00000 1.0000 1
Two-step selection 100 UNI_PEESE 1.00000 1.0000 1
Two-step selection 100 UNI_PET 1.00000 1.0000 1
Two-step selection 100 UNI_PET-PEESE 1.00000 1.0000 1
Two-step selection 100 UNI_RE 1.00000 1.0000 1
Two-step selection 100 UNI_RE-ISW 1.00000 1.0000 1
Two-step selection 100 UNI_WAAP 1.00000 1.0000 1
Two-step selection 100 UNI_WILS 1.00000 1.0000 1
Two-step selection 100 p-uniform 0.99998 0.9975 1
Two-step selection 100 p-uniform* 1.00000 1.0000 1

3 Univariate adjustment methods under one-step selection

3.1 Bias

3.1.1 \(\bar{\rho} = 0.2\)

3.1.1.1 \(k=10\)

3.1.1.2 \(k=30\)

3.1.1.3 \(k=60\)

3.1.1.4 \(k=100\)

3.1.2 \(\bar{\rho} = 0.4\)

3.1.2.1 \(k=10\)

3.1.2.2 \(k=30\)

3.1.2.3 \(k=60\)

3.1.2.4 \(k=100\)

3.1.3 \(\bar{\rho} = 0.8\)

3.1.3.1 \(k=10\)

3.1.3.2 \(k=30\)

3.1.3.3 \(k=60\)

3.1.3.4 \(k=100\)

3.2 RMSE

3.2.1 \(\bar{\rho} = 0.2\)

3.2.1.1 \(k=30\)

3.2.1.2 \(k=60\)

3.2.1.3 \(k=100\)

3.2.2 \(\bar{\rho} = 0.4\)

3.2.2.1 \(k=30\)

3.2.2.2 \(k=60\)

3.2.2.3 \(k=100\)

3.2.3 \(\bar{\rho} = 0.8\)

3.2.3.1 \(k=30\)

3.2.3.2 \(k=60\)

3.2.3.3 \(k=100\)

3.3 Confidence intervals

3.3.1 \(\bar{\rho} = 0.2\)

3.3.1.1 \(k=30\)

3.3.1.2 \(k=60\)

3.3.1.3 \(k=100\)

3.3.2 \(\bar{\rho} = 0.4\)

3.3.2.1 \(k=30\)

3.3.2.2 \(k=60\)

3.3.2.3 \(k=100\)

3.3.3 \(\bar{\rho} = 0.8\)

3.3.3.1 \(k=30\)

3.3.3.2 \(k=60\)

3.3.3.3 \(k=100\)

4 Comparison between one- and two-step selection for univariate adjustment methods

4.1 \(\bar{\rho} = 0.2\)

4.1.1 \(k=30\)

4.1.2 \(k=60\)

4.1.3 \(k=100\)

4.2 \(\bar{\rho} = 0.4\)

4.2.1 \(k=30\)

4.2.2 \(k=60\)

4.2.3 \(k=100\)

4.3 \(\bar{\rho} = 0.2\)

4.3.1 \(k=30\)

4.3.2 \(k=60\)

4.3.3 \(k=100\)

5 Novel adaptations performance

5.1 Comparsion between CHE-ISCW and CHE

5.1.1 \(\bar{\rho} = 0.2\)

5.1.1.1 \(k = 30\)

5.1.1.2 \(k = 60\)

5.1.1.3 \(k = 100\)

5.1.2 \(\bar{\rho} = 0.4\)

5.1.2.1 \(k = 30\)

5.1.2.2 \(k = 60\)

5.1.2.3 \(k = 100\)

5.1.3 \(\bar{\rho} = 0.8\)

5.1.3.1 \(k = 30\)

5.1.3.2 \(k = 60\)

5.1.3.3 \(k = 100\)

5.2 One-step selection

5.2.1 \(\bar{\rho} = 0.2\)

5.2.1.1 \(k = 30\)

5.2.1.2 \(k = 60\)

5.2.1.3 \(k = 100\)

5.2.2 \(\bar{\rho} = 0.4\)

5.2.2.1 \(k = 30\)

5.2.2.2 \(k = 60\)

5.2.2.3 \(k = 100\)

5.2.3 \(\bar{\rho} = 0.8\)

5.2.3.1 \(k = 30\)

5.2.3.2 \(k = 60\)

5.2.3.3 \(k = 100\)

5.3 Comparison between one- and two-step selection

5.3.1 \(\bar{\rho} = 0.2\)

5.3.1.1 \(k = 30\)

5.3.1.2 \(k = 60\)

5.3.1.3 \(k = 100\)

5.3.2 \(\bar{\rho} = 0.4\)

5.3.2.1 \(k = 30\)

5.3.2.2 \(k = 60\)

5.3.2.3 \(k = 100\)

5.3.3 \(\bar{\rho} = 0.8\)

5.3.3.1 \(k = 30\)

5.3.3.2 \(k = 60\)

5.3.3.3 \(k = 100\)

6 RQ3: Comparison of novel adaptations with univariate selection models

6.1 Bias

6.1.1 \(\bar{\rho} = 0.2\)

6.1.1.1 \(k = 10\)

6.1.1.2 \(k = 30\)

6.1.1.3 \(k = 60\)

6.1.1.4 \(k = 100\)

6.1.2 \(\bar{\rho} = 0.4\)

6.1.2.1 \(k = 10\)

6.1.2.2 \(k = 30\)

6.1.2.3 \(k = 60\)

6.1.2.4 \(k = 100\)

6.1.3 \(\bar{\rho} = 0.8\)

6.1.3.1 \(k = 10\)

6.1.3.2 \(k = 30\)

6.1.3.3 \(k = 60\)

6.1.3.4 \(k = 100\)

6.2 RMSE

6.2.1 \(\bar{\rho} = 0.2\)

6.2.1.1 \(k = 10\)

6.2.1.2 \(k = 30\)

6.2.1.3 \(k = 60\)

6.2.1.4 \(k = 100\)

6.2.2 \(\bar{\rho} = 0.4\)

6.2.2.1 \(k = 10\)

6.2.2.2 \(k = 30\)

6.2.2.3 \(k = 60\)

6.2.2.4 \(k = 100\)

6.2.3 \(\bar{\rho} = 0.8\)

6.2.3.1 \(k = 10\)

6.2.3.2 \(k = 30\)

6.2.3.3 \(k = 60\)

6.2.3.4 \(k = 100\)

6.3 Coverage

6.3.1 \(\bar{\rho} = 0.2\)

6.3.1.1 \(k = 10\)

6.3.1.2 \(k = 30\)

6.3.1.3 \(k = 60\)

6.3.1.4 \(k = 100\)

6.3.2 \(\bar{\rho} = 0.4\)

6.3.2.1 \(k = 10\)

6.3.2.2 \(k = 30\)

6.3.2.3 \(k = 60\)

6.3.2.4 \(k = 100\)

6.3.3 \(\bar{\rho} = 0.8\)

6.3.3.1 \(k = 10\)

6.3.3.2 \(k = 30\)

6.3.3.3 \(k = 60\)

6.3.3.4 \(k = 100\)

7 Compare un-adapted univariate methods

7.1 Overall

7.1.1 \(\bar{\rho} = 0.2\)

7.1.2 \(\bar{\rho} = 0.4\)

7.1.3 \(\bar{\rho} = 0.8\)

7.2 \(\bar{\rho} = 0.4\)

7.2.1 Bias

7.2.1.1 \(k = 10\)

The 3PSM, 4PSM, and p-uniform* are nearly unbiased. The p-uniform estimator has systematic increasing positive biases for smaller average effects and larger between-study heterogeneity.

7.2.1.2 \(k = 30\)

7.2.1.3 \(k = 60\)

7.2.1.4 \(k = 100\)

7.2.2 RMSE

7.2.2.1 \(k = 10\)

The 3PSM and p-uniform* are more accurate than 4PSM for small average effects and large between-study heterogeneity.

7.2.2.2 \(k = 30\)

7.2.2.3 \(k = 60\)

7.2.2.4 \(k = 100\)

7.2.3 Coverage

7.2.3.1 \(k = 10\)

The 3PSM and p-uniform* are more accurate than 4PSM for small average effects and large between-study heterogeneity.

7.2.3.2 \(k = 30\)

7.2.3.3 \(k = 60\)

7.2.3.4 \(k = 100\)